Heat the olive oil over medium-high heat. Add the sliced sausage and sauté until the edges are golden brown, about 4 minutes. Remove and set aside.
Melt the butter in the same skillet over medium heat. Add the onions and soften for 7 minutes, a little longer than usual to develop more flavor. Add the garlic and cook for 2 more minutes.
Add the chicken broth, mustard, brown sugar, hot sauce, and parsley and use a silicone spatula to “clean” the bottom and sides of the skillet, this will add more flavor to the dish.
Increase heat to medium-high. Add the cabbage and carrots and season with salt and pepper. Sauté for 10 to 15 minutes, or until desired texture is obtained.
Add the kielbasa back to the skillet and toss to combine and heat through.
Hillshire Farm Polska Kielbasa is my favorite kind of sausage for this recipe, but any kind of cased sausage may be used.
There is a hint of hot sauce in this recipe that doesn't make it spicy. It serves as a flavor enhancer along with mustard and a little brown sugar for a well-rounded flavor profile. I use Frank's Hot Sauce.